## Research Institute of Science & Technology

Research‎ > ‎

### Algebra/Algebraic Geometry

#### Projects

• Invariant theory
• Automorphisms of curves
• Field of moduli versus field of definition
• Theta functions and Jacobians
• Superelliptic curves

#### Theta Functions

Theta functions are some of the most amazing objects in mathematics.  They have been studied by Riemann, Picard, Kovalevski, Frobenious among many others.

We focus on theta functions of algebraic curves, and more explicitly on theta-nulls of superelliptic curves.

#### Genus 3 curves

Genus 3 curves are ternary quartics.  If the curve is hyperelliptic then they are written as
y^2=f(x)
where f(x) is a polynomial of degree 8.

Genus 3 hyperelliptic curves are treated in the package of hyperelliptic curves.

Here we deal with non-hyperelliptic genus 3 curves. The most famous of them all is perhaps the Klein's quartic.

#### AMS Special Session: March 2012, Tampa, Florida.

AMS Special Session:   Computational Algebraic Geometry and Applications
March 10-11, 2012, Tampa, Florida

Organizers:
• A. Elezi (aelezi@american.edu)

#### Genus 2 Curves

A Maple package for genus 2 curves which computes the basic invariants of the curve: Igusa invariants, absolute invariants, field of moduli, field of definition, automorphism group, etc.

#### Conferences and Activities

Algebraic Geometry Blog
• Theta Functions, Special Session, SIAM Conference, Raleigh, Oct. 6-8. 2011.
• Applied Algebraic Geometry, SIAM, Raleigh, Oct. 6-8. 2011.
• AMS sectional conference, Boise, Idaho, 2011
• AMS Annual Meeting, Boston, 2012

#### Databases for curves

I have seen every curve at least once ...

Genus: 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

#### Equation of curves over their field of definition

Let C be a curve defined over the complex numbers and F its field of definition.  Finding a curve \$X\$, isomorphic to C over the complex numbers, is an old problem in algebraic geometry.

Algorithms exist for genus 2 curves due to work of Clebsch, Mestre, Shaska, Cardona, et al.  We focus on genus 3 curves and on superelliptic curves of any genus.